Heron (also known as Hero) of Alexandria (fl.
A.D. 62) lived sometime between 100 B.C. and A.D. 100. Heron was
a Greek geometer, surveyor, and inventor who taught computational
geometry, arithmetic geodesy, physics, mechanics, and pneumatics
(theory and use of air pressure) at the Museum. Known as a rather
clever inventor, some of Heron's possible inventions include an
improved dioptra for surveyors, a screw cutter, a simple steam
engine (aeolipile), and war engines.

Heron's works exhibit a mixture of rigorous
mathematics and approximate procedures and formulas of the Egyptians.
He was also concerned with applied geometry and mechanics. His
works include *Metrica, Geometrica, Geodesy *(all give theorems
and rules for plane areas, surface areas, and volumes of several
figures)*, *and *Dioptra *(a book on land surveying)*.
*His applied works include *Mechanics, The Construction of
Catapults, Measurements, The Design of Guns, Pneumatica, On the
Art of Construction of Automata.* He gives designs for water
clocks, measuring instruments, automatic machines, weight-lifting
machines, and war engines.

Heron is most famous for his formula for the area of a triangle:

where *a, b, *and *c* are the lengths
of the triangle's sides and *s* is the **semiperimeter:**

Heron's proof relies on the basics of Euclidian
geometry. However, the proof seems to be random. It involves a
seemingly endless path of relationships, similarity, and ratios
that finally come together to make a proof. Heron begins with
some preliminary results. The first two come from Euclid.

** PROPOSITION 1**
The bisectors of the angles of a triangle meet at a point that
is the center of the triangle's inscribed circle. This point is
known as the incenter. (In the figure,

This is Proposition IV.4 of Euclid's

** PROPOSITION 2**
In a right triangle, if a perpendicular is drawn from the right
angle to the base, the triangles on each side of it are similar
to the whole triangle and to each other. (In the figure

This is Proposition VI.8 of Euclid's

** PROPOSITION 3**
In a right triangle, the midpoint of the hypotenuse is equidistant
from the three vertices. (In figure,

** PROPOSITION 4**
If

** PROPOSITION 5** The opposite angles of a cyclic quadrilateral
sum to two right angles.

This is Proposition III.22 of Euclid's

While these propositions seem irrelevant, they are what Heron used. Here is the proof that Heron used:

__PART A__

**
**Heron began by inscribing a circle
within the triangle. In the figure,

This gives us a link between the triangle's area and semiperimeter.

**
**In the figure

Since **ID, IE, **and **IF** are perpendicular
to the sides of the **D** **ABC**, then three pairs of congruent right triangles
are created:

Heron's segment **BG = s.** Thus,

(1) s c = BG AB

s c = AG.(2) s b

=BG AC

= (BD + AD + AG) (AF + CF)

= (BD + AD + CE) (AD + CE)

s b = BD.since AD = AF and AG = CE = CF. Likewise,

(3) s a = BG BC

= (BD + AD + AG) (BE + CE)

= (BD + AD + CE) (BD + CE)

s a = ADsince BD = BE and AG = CE.

As a result, the semiperimeter **s**, as
well as the quantities **(s a), (s b),*** *and

**(s - c) **all appear as segments in the diagram.

__PART C__

**
**Now Heron was left with the task of
linking these facts in such a way to complete his argument. The
figure

The result is quadrilateral

< AHB+< AIB= 2 right angles.

Now focus on the angles created by the angle bisectors and segments
**ID, IE, **and **IF.** By the congruences from **part
B**, we get three pairs of congruent angles, so that

2a + 2b + 2g= 4 right angles = 360 degrees

so

a + b + g= 2 right angles = 180 degrees.Since

(b + g)=m< AIB, then

a + < AIB= 2 right angles =< AHB + < AIB.

This seems rather insignificant, but will prove to be crucial to what Heron did next.

While Heron's proof uses elementary geometry,
it is quite intricate. It is hard to imagine how he must have
arrived at the directions that he chose to take! One fact of interest
is that an old Arabic manuscript written by Islamic scholar Abu'l
Raihan Muhammed al-Biruni centuries after Heron credits this result
to Archimedes. However, there are no Archimedean writings to support
this claim.

** Classics of Mathematics **(Ronald Callinger)

** Mathematical Thought from Ancient to
ModernTimes** (Morris Kline)

*Journey Through Genius*

**Britannica.com**